Algorithms for fast convolutions on motion groups (Q1582144)

The authors develop fast algorithms for the computation of convolution integrals on the motion group of the 3D Euclidean space (special Euclidean group \(\text{SE}(3)\)). Using irreducible unitary representations of \(\text{SE}(3)\) in operator form, the Fourier transform of functions on the motion group can be written as an integral over the product space \(\text{SE}(3)\otimes S^2\). The integral form of the Fourier transform matrix elements allows the application of FFT methods developed for \(\mathbb{R}^3\), \(S^2\) and \(\text{SO}(3)\). The authors provide an algorithm for the fast computation of Fourier transforms on \(\text{SE}(3)\) and discuss its complexity. Furthermore, they consider the Fourier transform for the 3D discrete motion group. The paper doesn't contain numerical results.